A034385 Expansion of (1-16*x)^(-1/4), related to quartic factorial numbers.

1, 4, 40, 480, 6240, 84864, 1188096, 16972800, 246105600, 3609548800, 53421322240, 796463349760, 11946950246400, 180123249868800, 2727580640870400, 41459225741230080, 632253192553758720

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..500

A. Straub, V. H. Moll, and T. Amdeberhan, The p-adic valuation of k-central binomial coefficients, Acta Arith. 140 (1) (2009) 31-41, eq (1.10).

Formula

a(n) = (4^n/n!)*A007696(n), n >= 1, a(0) := 1, A007696(n) = (4*n-3)!^4 := Product_{j = 1..n} 4*j - 3.

G.f.: (1 - 16*x)^(-1/4).

D-finite with recurrence: n*a(n) + 4*(-4*n + 3)*a(n-1) = 0. - R. J. Mathar, Jan 28 2020

From Peter Bala, Mar 31 2024: (Start)

a(n) = (-16)^n*binomial(-1/4, n).

a(n) ~ Gamma(3/4)/(sqrt(2)*Pi) * 16^n/n^(3/4).

E.g.f.: hypergeom([1/4], [1], 16*x).

a(n) = (16^n)*Sum_{k = 0..2*n} (-1)^k*binomial(-1/4, k)* binomial(-1/4, 2*n - k).

(16^n)*a(n) = Sum_{k = 0..2*n} (-1)^k*a(k)*a(2*n-k).

Sum_{k = 0..n} a(k)*a(n-k) = (4^n)*binomial(2*n, n) = A098430.

Sum_{k = 0..2*n} a(k)*a(2*n-k) = (16^n)*binomial(4*n, 2*n). (End)

Mathematica

CoefficientList[Series[1/Surd[1-16x, 4], {x, 0, 20}], x] (* Harvey P. Dale, Aug 06 2018 *)

Crossrefs

Cf. A007696.

Expansion of (1-b^2*x)^(-1/b): A000984 (b=2), A004987 (b=3), this sequence (b=4), A034688 (b=5), A004993 (b=6), A034835 (b=7), A034977 (b=8), A035024 (b=9), A035308 (b=10).

Sequence in context: A370444 A002705 A235372 * A249927 A218302 A296100

Adjacent sequences: A034382 A034383 A034384 * A034386 A034387 A034388

Keyword

nonn,easy,changed

Author

Wolfdieter Lang

Status

approved